The final version of the mean value theorem for harmonic functions
type: Item
instance of: scholarly article
title: The final version of the mean value theorem for harmonic functions
zbMATH Open document ID: 0894.31001
DOI: 10.1007/BF02308535
published in: Mathematical Notes
publication date: 1997-07-14T00:00:00Z
review text: Delsarte's two-circle theorem states that a locally integrable function \(f\) in \(\mathbb{R}^n\), \(n\geq 2\), is harmonic if it has the mean-value property on two spheres with radii \(r_1\) and \(r_2\) where \({r_1\over r_2}\) is not a ratio of roots of a special entire function \(n(z)\). A local version of the two-circle theorem [\textit{C. A. Berenstein} and \textit{R. Gay}, Isr. J. Math. 55, 267-388 (1986; Zbl 0624.31002)] considers the case when \(f\) is defined in a ball of radius \(R>r_1+r_2\). In this paper, the author investigates the case \(R\leq r_1+r_2\) to obtain the definitive version of the two-circle theorem.
reviewed by: Victor Anandam
Mathematics Subject Classification ID: 31B05
zbMATH DE Number: 1028516
zbMATH Keywords: mean-value property
zbMATH Keywords: two-circle theorem
author: Valeriĭ V. Volchkov
MaRDI profile type: Publication
cites work: The converse of Gauß's theorem for harmonic functions
cites work: Offbeat Integral Geometry
cites work: Mean values and differential equations
cites work: Pompeiu's problem on symmetric spaces
cites work: A local version of the two-circles theorem
cites work: Morera type theorems on the unit disc
cites work: Q4288961
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cites work: Q5640160
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full work available at URL: https://doi.org/10.1007/bf02308535
OpenAlex ID: W2084848357
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description: scientific article; zbMATH DE number 1028516